×

On convex and \(*\)-concave multifunctions. (English) Zbl 1085.39024

A characterization of *-concave and convex multifunctions which are continuous with respect to the Hausdorff metric is given. Namely, {a continuous multifunction} \(F:[a,b] \to \text{ clb}(Y)\) is *-concave (resp. convex) if and only if for every \(s,t \in [a,b], \; s<t\) the inclusion \[ \frac{1}{t-s} \int_{s}^{t} F(x)dx \subset (\text{resp.} \supset) \frac{F(s) \overset{*}{+} F(t)}{2} \] holds.
The main tool in the proof is the characterization of real convex function which is based upon Hadamard inequality.

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
26E25 Set-valued functions
PDF BibTeX XML Cite
Full Text: DOI